Test of the generalized Brink-Axel hypothesis in 64,65Ni

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Test of the generalized Brink-Axel hypothesis in

^64,65

Ni

L. Crespo Campo,^1,*M. Guttormsen,¹F. L. Bello Garrote,¹T. K. Eriksen,²F. Giacoppo,^3,4A. Görgen,¹K. Hadynska-Klek,⁵ M. Klintefjord,¹A. C. Larsen,¹T. Renstrøm,¹E. Sahin,¹S. Siem,¹A. Springer,⁶T. G. Tornyi,²and G. M. Tveten¹

1Department of Physics, University of Oslo, N-0316 Oslo, Norway

2Department of Nuclear Physics, Research School of Physics and Engineering, The Australian National University, Acton, Australia Capital Territory 2601, Australia

3Helmholtz Institute Mainz, 55099 Mainz, Germany

4GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany

5Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Legnaro, 2 35020 Legnaro (Padua), Italy

6Department of Physics, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany

(Received 23 June 2017; revised manuscript received 18 February 2018; published 6 November 2018) Previously published particle-γ coincidence data on the⁶⁴Ni(p, p^′γ)⁶⁴Ni and⁶⁴Ni(d, pγ)⁶⁵Ni reactions were further analyzed to study the statistical properties of γ decay in ^64,65Ni. To do so, the γ decay to the quasicontinuum region and discrete low-lying states was investigated at γ-ray energies of 2.0–9.6 and 1.6–6.1 MeV in⁶⁴Ni and⁶⁵Ni, respectively. In particular, the dependence of theγ-strength function with initial and final excitation energy was studied to test the validity of the generalized Brink-Axel hypothesis. Finally, the role of fluctuations in transition strengths was estimated as a function of γ-ray and excitation energy.

Theγ-strength function is consistent with the hypothesis of the independence of initial excitation energy, in accordance with the generalized Brink-Axel hypothesis. The results show that theγ decay to low-lying levels displays large fluctuations for low initial excitation energies.

DOI:10.1103/PhysRevC.98.054303

I. INTRODUCTION

Many analytical techniques in nuclear physics are based on the hypothesis of nuclearγ decay from a compound nuclear state: it is assumed that a given reaction has led to a compound state which decays independently of how it was formed [1]. At sufficiently high excitation energies, the nucleus is considered to be in its quasicontinuum regime, a region where the number of nuclear states is so high that they mix strongly with each other. At that point, nuclearγdecay is studied as an average of a large number of transitions and thus two statistical properties can be defined: the nuclear level density (NLD), or number of nuclear states per unit of energy, and theγ-strength function (γSF), or average reduced γ-ray transition probability [2].

Furthermore, the calculations are often simplified by assuming that the generalized Brink-Axel (gBA) hypothesis is valid.

In general terms, the gBA hypothesis implies that the dipole γ strength is independent of the structure of the initial state;

i.e, it has no explicit dependence on the excitation energy, spin, or parity, except for the obvious selection rules for dipole transitions [3,4].

When the gBA hypothesis holds, theγSF depends solely on theγ-ray energy for dipole radiation [2]. The gBA hypothesis simplifies the description of nuclearγ andβ decays, being frequently used in the calculation of neutron-capture cross sections [5–8]. Determining the circumstances under which the gBA hypothesis holds is therefore of great importance, due

*[email protected]

to its impact in fields such as nuclear astrophysics and reactor physics.

The experimental and theoretical attempts made to validate the gBA hypothesis demonstrate that it is a question of great complexity. Some of the first verifications of this hypothesis below nucleon-emission thresholds come from the analysis of mainly E1 high-energy primary transitions, either from isolated resonances or from average neutron-capture data [9].

In addition, the observations ofM1 scissors modes in¹⁶³Dy [10] and¹⁷²Yb [11], deduced from radiative neutron-capture measurements, also support this hypothesis. Furthermore, the results from Refs. [12,13] show that, for a heavy odd-odd nucleus such as²³⁸Np, the NLD is extremely high and averaging over manyγ transitions can be performed. Under these circumstances, there is a significant suppression of Porter- Thomas [14] fluctuations, i.e, fluctuations in the individual partial radiative widths related to the complexity of the nuclear states involved in the decay. Thus, the gBA hypothesis could be reliably tested in ²³⁸Np. For lighter nuclei, however, the situation is normally rather different and large Porter-Thomas fluctuations are present [13,15]. Such fluctuations can be un- derstood in terms of the wave functions of the various nuclear states involved in a given decay. The reduced width amplitude depends on the wave functions of both the compound nuclear state and the exit channel. If these wave functions are of great complexity, the statistical model can be applied: the strengths of γ-ray transitions (or rather the partial radiative widths) are assumed to follow a Porter-Thomas distribution and the various exit channels are assumed to be independent [14]. For sufficiently heavy nuclei and/or when high excitation energies

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are involved, the complexity of the nuclear wave functions allows for the application of such a statistical model hypothesis. However, for light nuclei the lower NLD (and lower complexity of the involved wave functions) can challenge these statistical descriptions and result in large fluctuations of the radiative widths [16], thus making the validation of the gBA hypothesis difficult.

Despite the continued interest in the gBA hypothesis, the available information regarding this matter is still quite scarce.

More experimental data on γ-decay properties would help in understanding the connection between the validity of this hypothesis and nuclear structure phenomena. Furthermore, these measurements could greatly improve the estimation of (n,γ) cross sections, which are often highly uncertain for unstable neutron-rich nuclei [17,18].

In this work, previously published particle-γ coincidence data on the⁶⁴Ni(p, p^′γ)⁶⁴Ni and⁶⁴Ni(d, pγ)⁶⁵Ni reactions from Refs. [19,20] were further analyzed to study the dependence of theγSF with initial and final excitation energy.

In other words, we investigated the energy regions under which the use of an excitation-energy-independent γSF is appropriate and thus tested the validity of the gBA hypothesis.

This article is organized as follows: In Sec. IIthe experimental details and analytical method are summarized and the new results presented. The γSF for different excitation energies is studied in Secs. II A and II B. In Sec. III the fluctuations of theγSF are discussed. Finally, a summary is given in Sec.IV.

II. THEγSF OF^64,65Ni AND ITS DEPENDENCE ON EXCITATION ENERGY

A. The standard Oslo method applied to various energy regions The data employed in this work were first presented in Refs. [19,20], where details on the experimental method are given. As shown there, particle-γ coincidences were measured and used to obtain coincidence matrices for ⁶⁴Ni and

65Ni, in which theγ rays emitted from a given excitation energy are represented. Applying the Oslo method [21–24], the

spectra were unfolded [21] and the first-generation (primary) γ rays were extracted [22].

A section in the primary matrix was selected to extract data corresponding to statistical γ decay from compound nuclear states, for which the gBA hypothesis was expected to hold [3,25]. Then, the selected region in the primary matrix was used to obtain the functional form of the NLDρ[26] and theγ-transmission coefficient [2]T through

P(Eγ, Ei)∝ρ(Ef)T(Eγ), (1) where P(Eγ, Ei) represents the probability of γ decay from an initial excitation energy E_i with a γ-ray energy E_γ, obtained from the primary coincidence matrix U as P(Eγ, Ei)=U(Eγ, Ei)/!

Eγ,U(Eγ, Ei).Efis the final excitation energy, withEf =Ei−Eγ. FromT, theγSF was obtained with the relation [27,28]

fL(Eγ)= T(Eγ)

2πE^2L_γ ⁺¹, (2) where L is the multipolarity of the transition, here taken as L=1 since dipole radiation is expected to represent the main contribution to theγSF in the quasicontinuum [27,29].

The NLD and γSF were then normalized using additional experimental data.

The primary coincidence matrices for^64,65Ni obtained in Refs. [19,20] are included in Figs.1(a)and2(a). The regions used for the extraction of the NLD andγSF are also shown for the standard Oslo method. For ⁶⁴Ni, the region corresponds toE_γ >1.98 MeV and E_x=5.82–9.66 MeV [19], while for ⁶⁵Ni the chosen limits were E_γ >1.60 MeV and Ex=4.43–6.08 MeV [20]. The resultingγSFs are shown in Figs.1(b)and2(b). The excitation energy resolution (FWHM) is ≈130 keV at 692 keV and it is approximately constant with excitation energy, while theγ-ray energy resolution is

≈70 keV at 1017 keV and scales with"

E_γ.

As seen in Figs.1 and2, the selected region contains a large area for which the data vary smoothly with excitation and γ energy and thus the decay from compound nuclear states is expected to dominate. Nevertheless, strong diagonals

-ray energy (MeV)

2 γ 4 6 8 10

(MeV)xE

0 2 4 6 8 10

# of counts

1 10 102

103

No. of counts

1 10 102

103

(a)

2 1

1 2 3γ 4 5 6 7 8 9

)-3 SF (MeVγ

−8

10

−7

10

Total Region 1 Region 2

(b)

64

Ni

FIG. 1. TheγSF as obtained from different regions in the first-generation coincidence matrix for⁶⁴Ni: (a) the considered regions in the coincidence matrix and (b) the resultingγSF. The total region corresponds to the standardγSF as obtained in Ref. [19].

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1 γ2 3 4 5 6 7

(MeV)xE

0 1 2 3 4 5 6 7

# of counts

1 10 102

No. of counts

1 10 102

(a)

2 1

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6γ )-3 SF (MeVγ

8

10− 7

10−

Total Region 1 Region 2

65

Ni

(b)

FIG. 2. TheγSF obtained from different regions in the first-generation coincidence matrix for⁶⁵Ni [20]: (a) the considered regions in the coincidence matrix and (b) the resultingγSF. The total region corresponds to the standardγSF.

are present. In the case of ⁶⁴Ni, a diagonal at Ex ≈Eγ is seen, corresponding to the decay to the ground state. A much stronger feeding to the first excited 2⁺ state is measured, corresponding to the diagonal atEx ≈Eγ −1.345 MeV [19].

For⁶⁵Ni, the most pronounced diagonal is observed atEx ≈ Eγ, indicating the presence of strongγ-ray transitions feeding both the ground state 5/2⁻and the first excited 1/2⁻isomeric state at 63.37 keV. Since the neutron separation energy Sn

for ⁶⁵Ni is considerably lower (Sn=6.098 MeV) than for

64Ni, theseγ-decay transitions are strong even at excitation energies very close toSn[20].

To study the impact of these transitions in theγSFs of

64,65Ni, the Oslo method was applied to smaller and smoother regions of the first-generation coincidence matrix. This technique was used for⁶⁵Ni in Ref. [20] as shown in Fig.2. In this work, the same technique was applied to⁶⁴Ni and the results are included in Fig.1. The regions used for the analysis are depicted in Figs.1(a)and2(a), while the resultingγSFs are shown in Figs.1(b)and2(b). The results are compared to the standardγSF obtained in Refs. [19,20].

For both⁶⁴Ni and ⁶⁵Ni, the strong diagonal at E_x ≈E_γ was excluded in region 1. Furthermore, a smaller section of the matrix (region 2) was selected and the corresponding γSF included. In the case of⁶⁴Ni, region 2 is obtained when excluding the strong diagonal that corresponds to the feeding of the first excited 2⁺state at 1.345 MeV.

As seen in Fig.1for⁶⁴Ni, theγSFs obtained from regions 1 and 2 are in good agreement with the standard γSF at γ-ray energies below≈6.0 MeV, although some deviations are seen at Eγ ≈5–6.0 MeV. At Eγ ≈6.0–7.6 MeV, the results from region 1 are substantially higher than the standard ones. Based solely on the results from Fig.1, this could be interpreted as being partially due to the strong feeding of the first excited state: in the standardγSF, the results with Eγ >6.0 MeV include γ decays to both the ground and the first excited 2⁺ states, while in the results from region 1 the decay to the ground state is excluded. As seen from the coincidence matrix, the feeding of the first excited state is stronger than for the ground state, and thus this could be

seen as being due to a standard γSF which is lower than for region 1 aboveE_γ ≈6.0 MeV. However, the coincidence matrix contains contributions from both the NLD and theγSF and, as shown later in Fig.7, theγSF feeding the first excited state is perfectly compatible with the standard γSF, at least for Ei >6 MeV. This leads us to the conclusion that the discrepancies observed in Fig.1(b)between theγSFs atE_γ >

6 MeV are most likely due to differences in the normalizations for the various regions, which cannot be exactly the same with this particular procedure. Furthermore, differences in the normalization are more likely to impact the resulting γSFs atγ-ray energies approachingE_γ ≈S_n−E_γ,lowand atE_γ >

S_n−E_γ,low. This is due to the applied extrapolation at such energies, whereE_γ,lowis the minimum value ofEγ used in the extraction of theγSF via the Oslo method [19,24]. The results presented in Fig.1suggest that theγSF for⁶⁴Ni obtained in Ref. [19] (the standard γSF) is indeed a good estimate for E_γ <6.0 MeV. Above that energy, the standardγSF is most likely to be the best estimate as well, extracted using all the available data (total region in Fig.1).

As shown in Fig.2 the results for ⁶⁵Ni are in very good agreement with the standard γSF [20]. The low-energy enhancement or upbend¹is reproduced and the analysis confirms that the resonancelike structure at Eγ =4.6 MeV is truly present and that it is not just the result of a strong feeding to the ground state.

1The expressions low-energy enhancement and upbend are here used to describe an increase in the γSF with decreasing γ-ray energy at energies belowEγ≈2–4 MeV. An enhancement is observed below Eγ ≈3 MeV for^64,65Ni [19,20] when compared to the descriptions given by Lorentzian models such as the standard Lorentzian [27], which predicts a decrease in the γSF at lowγ- ray energies. The low-energy enhancement is also observed when compared to the predictions from the generalized Lorentzian for

64Ni, as shown in Ref. [19].

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B. TheγSF as a function of initial and final states To test the validity of the gBA hypothesis, the dependence of theγSF with both initial and final excitation energy was studied applying the technique described in Ref. [12]. Assum- ing that theγ-transmission coefficientT only depends onE_γ, we introduce a normalization factorNwhich only depends on the initial excitation energy and rewrite Eq. (1) as

N(Ei)P(Eγ, Ei)=ρ(Ei−Eγ)T(Eγ), (3) which determines the normalization factor by

N(Ei)=

#Ei

0 T(Eγ)ρ(Ei−Eγ)dEγ

#E_i

0 P(Eγ, E_i)dEγ

. (4)

Since ρ is known, T can be studied in detail for each excitation energy bin byN P /ρ, as given in Eq. (3). Therefore, for the initial excitation energy states we obtain [12]

T(Eγ, Ei)=N(Ei)P(Eγ, E_i)

ρ(Ei−E_γ) . (5) In a similar way, we defineT(Eγ, Ef) and study the decay to final excitation energies:

T(Eγ, Ef)= N(Ef+E_γ)P(Eγ, E_f +E_γ) ρ(Ef) . (6) The normalization factorN is obtained assuming that both T(Eγ, Ei) andT(Eγ, Ef) fluctuate around the excitation- energy-independent T(Eγ) obtained with the standard Oslo method from Eqs. (1) and (4). Applying the relation between theγSF andT given by Eq. (2), the correspondingγSF functions for the initial and final states,f(Eγ, E_i) andf(Eγ, E_f), were obtained and averaged over the initial and final excitation energies by [12]

fi(Eγ)= 1 S_n−E_γ

$ S_n E_γ

f(Eγ, Ei)dEi, (7) ff(Eγ)= 1

S_n−E_γ

$ S_n−E_γ 0

f(Eγ, Ef)dEf, (8) where Eγ >1.98 MeV andSn=9.66 MeV for ⁶⁴Ni while Eγ >1.60 MeV andSn=6.08 MeV for⁶⁵Ni [19,20,30].

To check that the normalization functionN(Ei) is reason- able, the results for fi(Eγ),ff(Eγ), and theγSFs obtained with the standard Oslo method f(Eγ) were compared, as shown in Figs.3and4. The functions are in good agreement, supporting the applied normalization function.

With the well-behaving normalization function, the γSF was studied as a function of initial and final excitation energies. The dependence of theγSF with initial excitation energy f(Eγ, E_i) is shown for^64,65Ni in Figs.5and6, respectively.

Note that eachf(Eγ, Ei) is built on a given initial excitation- energy gate, but with no specific final state. However, for a given Eγ andEi, the final excitation energy is determined asEf =Ei−Eγ. TheγSFs obtained with the standard Oslo method,f(Eγ), are also shown in Figs.5and6(solid line).

The agreement between the obtainedf(Eγ, E_i) and the standard strength is clear for both ^64,65Ni. The uncertainties are, however, large for ⁶⁵Ni, due to lower statistics. The results

-ray energy (MeV) γ

2 3 4 5 6 7 8 9 10

)

-3

SF (MeV γ

−8

10

−7

10

Eq. (1) standard Eq. (7) initial Eq. (8) final

FIG. 3. Comparison off(Eγ),fi(Eγ), andff(Eγ) for⁶⁴Ni.

forf(Eγ, Ei) reproduce the structures present in the standard γSF for both nuclei within uncertainties. Not only is the measured low-energy enhancement present for the variousEi, but its shape seems independent ofE_ifor both^64,65Ni.

The resemblance between the individualf(Eγ, E_i) with the standardf(Eγ) is due to the averaging over a relatively high number of transitions, thus preventing large fluctuations. However, exploiting the decay to a few final states will enhance such fluctuations dramatically. Figure 7 shows the average γSF feeding a given final excitation energy bin for⁶⁴Ni,f(Eγ, Ef). The corresponding results for⁶⁵Ni are shown in Fig. 8. In both cases the Ef bins were chosen to

-ray energy (MeV)

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 γ )

-3

SF (MeV γ

−8

10

−7

10

Eq. (1) standard Eq. (7) initial Eq. (8) final

FIG. 4. Comparison off(Eγ),fi(Eγ), andff(Eγ) for⁶⁵Ni.

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) -3 SF (MeVγ

9

10− 8

10− 7

10−

= 9.50 MeV (a) E i

Data Standard

1 2 3 4 5 6 7 8 9 )-3 SF (MeVγ

9

10− 8

10− 7

10−

= 8.10 MeV (d) E i

= 9.20 MeV (b) E i

-ray energy (MeV) 1 2 3 4 5 6 7 8 9γ

= 7.70 MeV (e) E i

= 8.50 MeV (c) E i

1 2 3 4 5 6 7 8 9 = 6.70 MeV

(f) E i

FIG. 5. γ strength from several initial excitation energies in⁶⁴Ni. The results are compared to the standardγSF (solid line). Each initial excitation energy bin has a width of 124 keV.

contain discrete energy levels, with widths of 124 and 248 keV for ⁶⁴Ni and ⁶⁵Ni, respectively. It is important to note that the standard γSF, f(Eγ), is obtained at excitation energies aboveE_x,min=5.82 and 4.43 MeV for⁶⁴Ni and⁶⁵Ni. At those excitation energies, theγSF was expected to be independent of excitation energy [19,20]. Therefore, for a givenEf, the results for f(Eγ, Ef) with Eγ +Ef > E_x,min MeV should be in agreement with the standardγSF,f(Eγ). As observed in Figs. 7 and 8, this is the case for both ⁶⁴Ni and ⁶⁵Ni, considering the presence of statistical uncertainties and fluctuations. Only two data points in Figs.7(a)and7(b)show larger deviations with respect to f(Eγ), corresponding to decays fromEi ≈6.2–6.4 MeV. A vertical dashed line is included in Figs.7and8to indicate the limitEγ =E_x,min−Ef.

In the case of⁶⁴Ni, the strengths feeding states aboveEf ≈ 3.3 MeV are in very good agreement with the standardγSF, as seen in Figs.7(e)and7(f). A good agreement is also seen forE_f =2.972 MeV atE_γ >2.8 MeV. The feeding of the 2⁺ state at 2.276 MeV [30] also shows a similar trend as the standardγSF aboveEγ ≈3.5 MeV, although significant deviations are observed. In other words, the results forf(Eγ, Ef)

in Figs.7(c)and7(d)are in agreement withf(Eγ) forγ-ray energies that correspond to Ei >5.6–5.8 MeV, close to the lower limit of theE_i range used for the extraction off(Eγ).

This confirms that an excitation-energy-independentγSF is a good assumption aboveEi ≈5.8 MeV. As seen in Fig.7(c), belowEγ ≈3.5 MeV some fluctuations are present, although they are scattered around the standardγSF. Therefore, overall the results from Fig. 7 suggest that the γSF is independent of E_f at and above ≈3.0 MeV. The strength feeding the ground and first excited states presents very large fluctuations, which are scattered around the standardγSF. The presence of fluctuations is to be expected given the strong Porter-Thomas fluctuations. For instance, at Ei =3.3–4 MeV, only three transitions to the ground state have been seen according to Ref. [30]. As a result, fluctuations of an order of magnitude are seen belowEγ ≈5 MeV in Fig.7(a). Furthermore, deviations from the standardγSF are also expected due to the presence of quadrupole transitions. It should be noted that⁶⁴Ni has many excited states with spinI =2 belowE_x ≈4 MeV, which can decay to the ground state with quadrupole radiation. Since the γSF is extracted assuming the emission of dipole radiation

1 2 3 4 5 6

)-3 SF (MeVγ

−9

10

8

10−

−7

10

6

10−

= 4.8 MeV (a) E i

1 2γ 3 4 5 6

= 5.2 MeV (b) E i

1 2 3 4 5 6

Data Standard = 6.0 MeV

(c) E i

FIG. 6. γ strength from several initial excitation energies in⁶⁴Ni. The results are compared to the standardγSF (solid line). Each initial excitation energy bin has a width of 248 keV.

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)-3 SF (MeVγ ⁸⁻10

−7

10

= 0.0 MeV at Ef

(a) 0+

1 2 3 4 5 6 7 8 9

)-3 SF (MeVγ ⁸⁻10

−7

10

= 2.972 MeV ) at E f

(d) (1,2+

= 1.345 MeV at E f

(b) 2+

-ray energy (MeV) 1 2 3 4 5 6 7 8 9γ

= 3.340 MeV (e) E f

= 2.276 MeV at E f

(c) (2)+

1 2 3 4 5 6 7 8 9 = 4.770 MeV

(f) E f

FIG. 7. γstrength feeding different final excitation energies for⁶⁴Ni. The results are compared to the standardγSF obtained as detailed in Sec.II(solid line). Each final excitation energy bin has a width of 124 keV. The vertical dashed line marks the limitEγ=Ex,min−Ef, where E_x,minis the minimum excitation energy used in the extraction of the standardγSF, hereE_x,min=5.82 MeV.

in Eq. (2), the estimated strength for a quadrupole transition would be off by a factor ofE_γ².

In the case of⁶⁵Ni, the results forf(Eγ, Ef) are in agreement with the standardγSF forEf at and above 1.92 MeV.

Above the vertical dashed line, a good agreement is seen for all values of E_f considered. Furthermore, even though significant deviations fromf(Eγ) are seen atEf =0.69 MeV below the vertical dashed line, the data points are approximately scattered aroundf(Eγ). As seen in Figs.8(a)–8(c), the resonance-like structure present in the standardγSF atE_γ ≈ 4.6 MeV is also observed inf(Eγ, E_f) for the differentE_f bins considered. The low-energy enhancement is well reproduced, as seen in Figs.8(e)and8(f). BelowEf =0.69 MeV, larger deviations are observed. Again, a contributing factor to these deviations could be the presence of quadrupole transitions. The results shown in Fig. 8(a) suggest that the ground state is less strongly fed from excitation energies

below≈4 MeV, wheref(Eγ, Ef) is below f(Eγ), while in Fig.8(b)the strength populating the second excited state at E_f =0.31 MeV is abovef(Eγ) atE_γ ≈2–3 MeV. However, the average strength feeding the ground, first, and second excited states is in agreement with the standard γSF. Note that, as previously shown in Fig.2, the upbend and resonancelike structure atEγ ≈4.6 MeV are seen even when the decay to those states is excluded. In conclusion, the overall agreement between the standard γSF and the variousf(Eγ, Ef) shown in Fig.8 indicates that no clear dependence on final excitation energy is observed in theγSF of⁶⁵Ni aboveEf ≈ 0.69 MeV.

III. FLUCTUATIONS OF THEγSF

Validating the gBA hypothesis becomes especially difficult when strong Porter-Thomas fluctuations are present, as is

)-3SF (MeVγ

−9

10

−8

10

−7

10

= 0.0,0.069 MeV at E f

, 1/2-

(a) 5/2-

1 2 3 4 5 6

)-3SF (MeVγ

−9

10

−8

10

−7

10

= 1.01 MeV at E f

(d) 9/2+

= 0.31 MeV at E f

(b) 3/2-

1 2γ 3 4 5 6

= 1.92,2.14 MeV at E f

, 3/2-

(e) 5/2+

= 0.69 MeV at E f

(c) 3/2-

1 2 3 4 5 6

= 2.9 MeV at E f

, 5/2+

(f) 3/2+

FIG. 8. γ strength feeding different final excitation energies for⁶⁵Ni. The results are compared to the standardγSF (solid line). Each final excitation energy bin has a width of 248 keV. The vertical dashed line marks the limitEγ=Ex,min−Ef, whereEx,minis the minimum excitation energy used in the extraction of the standardγSF, hereEx,min=4.43 MeV.

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) γr( E

2

10− 1

10−

1 E_i = 7.7 MeV

= 8.5 MeV Ei

= 9.5 MeV Ei

= 5.82 - 9.66 MeV Ei

64

Ni

(a)

-ray energy (MeV) 2.5 3 3.5 4γ 4.5 5 5.5 6 )γr( E

−2

10

1

10−

1

(b) = 3.3 MeV Ef

= 4.0 MeV Ef

= 4.8 MeV Ef

FIG. 9. Fluctuationsr(Eγ) in⁶⁴Ni, with r=√2/n: (a) the results for a specific initial energy bin, obtained withn=n(Ei, Eγ) as given by Eq. (10), and (b) the equivalent results for a final excitation energy bin. The results are compared tor(Eγ) for the standardγSF, obtained for a wider range of initial excitation energies.

often the case for light and medium mass nuclei. In^64,65Ni, the neutron separation energies are Sn=9.658 MeV and Sn=6.098 MeV, respectively. At these energies, the NLD is ≈2600 levels/MeV for ⁶⁴Ni and≈1100 levels/MeV for

65Ni. In contrast, heavier nuclei such as²³⁸Np present a much larger number of accessible nuclear states, with a NLD of 43 million levels/MeV at Sn=5.488 MeV. As shown in Ref. [13], Porter-Thomas fluctuations are not significant for

238Np, but they are expected to be larger for lighter nuclei. It is therefore important to estimate more quantitatively if this is the case for^64,65Ni before extracting conclusions regarding the validity of the gBA in these nuclei.

Following the procedure described in Ref. [13], we here assume that the fluctuations in theγSF follow the χ_ν² distribution, withνequal to the number of transitions,n, included in the averaging for a specificE_γ. For aχ_ν² distribution we define the ratior between the deviationσ and the averageµ asr =σ/µ=√2/ν. Using the experimental results for the NLD of^64,65Ni, we count the number of expected transitions from an initial to a final excitation energy bin n(Eγ), and withν=n(Eγ) we obtainr(Eγ)="

2/n(Eγ). To improve the averaging, a large region of the primary matrix was used as in the Oslo method. The number of transitions was

) γr(E

1

10−

1

= 4.8 MeV Ei

= 5.2 MeV Ei

= 6.0 MeV Ei

= 4.43 - 6.00 MeV Ei

65

Ni

(a)

-ray energy (MeV) 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6γ ) γr( E

−1

10 1

(b) = 2.1 MeV Ef

= 2.5 MeV Ef

= 3.0 MeV Ef

FIG. 10. Fluctuationsr(Eγ) in⁶⁵Ni, withr=√2/n. See text in Fig.9.

estimated as

n(Eγ)=!E²

E%i,max

E_i=Ei,min

%

Iπ

%1 δ=−1

%

π′

ρ(Ei, I,π)ρ

×(Ei−E_γ, I+δ,π^′), (9) where!Eis the bin width for the excitation energy,I andπ are the spin and parity of the initial state, andπ^′is the parity of the final state. Note that the sum overδruns from−1 to+1, corresponding to dipole radiation. For a decay from a specific Eibin, the number of transitions is

n(Eγ, Ei)=!E²%

Iπ

%1 δ=−1

%

π^′

ρ(Ei, I,π)ρ

×(Ei−Eγ, I+δ,π^′). (10) In a similar way, the number of transitions feeding a given final excitation energy binEf =Ei−Eγ was obtained.

Figures 9 and 10 show the expected fluctuations in the γ decay of ^64,65Ni as a function of γ-ray energy r(Eγ).

Figures9(a)and10(a)display the fluctuations from a given Ei bin in comparison with the results for the standard Oslo method, obtained forEi=5.82–9.66 MeV for⁶⁴Ni andEi = 4.43–6.0 MeV for ⁶⁵Ni, as indicated in Sec. II A. To study the fluctuations within the quasicontinuum, the decays for a givenE_ihere studied correspond toE_f >3 MeV. In addition, the analysis of the fluctuations in the feeding of a givenEf

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2.5 3 3.5 4 4.5 5 )γ) / f(E i , Eγ Ratio f(E

1

= 7.7 MeV (a) Ei

2.5 3γ 3.5 4 4.5 5 5.5 6 = 8.5 MeV

(b) E i

2.5 3 3.5 4 4.5 5 5.5 6 6.5

1

= 9.5 MeV (c) E i

FIG. 11. Ratio off(Eγ, Ei) for⁶⁴Ni with respect to the excitation-energy-independentγSF,f(Eγ). The results are shown forEi=7.7, 8.5, and 9.5 MeV.

bin is shown in Figs. 9(b)and10(b). The bin widths for the excitation energies are 124 and 248 keV for⁶⁴Ni and ⁶⁵Ni, respectively.

For a given initial excitation energy, r(Eγ) increases exponentially withEγ for both⁶⁴Ni and⁶⁵Ni. Note that, forEi

close to S_n,r(Eγ) ranges from≈0.01 to≈0.10 in⁶⁴Ni and from≈0.06 to≈0.17 in⁶⁵Ni. As the initial excitation energy decreases, fewer nuclear states are present and therefore fewer transitions are observed, resulting in larger fluctuations. For example, r(Eγ) for⁶⁴Ni fromEi =7.7 MeV is about four times higher than fromEi =9.5 MeV for a givenEγ.

The results for the standard γSF obtained for a wider range of initial excitation energies contain a better averaging over transitions and therefore much smaller fluctuations. For instance,r(Eγ) for⁶⁵Ni obtained forEi =4.43−6.00 MeV (the Ei range for the standard γSF) is about 8 times lower than ther(Eγ) obtained when only transitions from theEi= 4.8 MeV bin are considered.

As seen in Figs.9(b)and10(b), the fluctuations decrease exponentially withEγ and, as expected, they increase asEf

decreases, since fewer states are available and therefore fewer transitions take place. For instance, in⁶⁴Ni atEf =3.3 MeV, the value ofr(Eγ) is about three times larger than for decays toE_f =4.8 MeV for a givenE_γ.

In addition, the fluctuations in the γSF were studied by comparing the results for the γ strength for a given initial and final excitation energy f(Eγ, Ei) and f(Eγ, Ef) with

theEx-independentγSF,f(Eγ) [15]. In particular, the ratios of f(Eγ, Ei) and f(Eγ, Ef) to the average f(Eγ) were obtained:

R(Eγ, Ex)=f(Eγ, Ex)

f(Eγ) , (11) where x =i, f for initial or final excitation energies. The analysis was done for the⁶⁴Ni data, with better statistics. First, the fluctuations were studied fromR(Eγ, Ei) withEi =7.7, 8.5, and 9.5 MeV, where the level density is ≈500, 800, and 1200 levels/MeV, respectively. To analyze the impact of fluctuations in decays within the quasicontinuum, only decays withEf =Ei+Eγ >3 MeV were included (withρ(Ef)>

10 levels/MeV). The results, shown in Fig. 11, are clearly scattered aroundR(Eγ, E_i)=1, with the data points mostly contained in an≈15% band aroundR(Eγ, E_i)=1. When all data points are considered, the average ratio is 1.03. Further- more, for allEi considered, the deviations fromR(Eγ, Ei)= 1 are less than 7% on average. In other words, the results are consistent with the excitation-energy-independentγSF within

≈7% and the deviations can be interpreted as remnants of Porter-Thomas fluctuations, in agreement with the observations for⁴⁶Ti in Ref. [15].

To study the impact of decays to low-lying states in the observed fluctuations, the ratioR(Eγ, Ef) was obtained for E_f =0, 1.345, and 3.340 MeV and the results are shown in Fig.12. WithE_f =0 MeV andE_γ =E_i >6.4 MeV the ratio

3 4 5 6 7 8 9

) γ) / f(E f , E γ Ratio f(E

−1

10 1

= 0.0 MeV at Ef

(a) 0+

3 γ4 5 6 7 8

= 1.345 MeV at E f

(b) 2+

2.5 3 3.5 4 4.5 5 5.5 6 6.5 = 3.340 MeV

(c) E f

FIG. 12. Ratio off(Eγ, Ef) for⁶⁴Ni with respect to the excitation-energy-independentγSF,f(Eγ). The results are shown forEf =0, 1.35, and 3.34 MeV.

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ranges from 0.71 to 1.28, with an average of≈0.94, but its fluctuations increase drastically forEγ =Ei <6.4 MeV. In other words, even though only one state is available in the final excitation energy binE_f =0 MeV, a ratio close to 1 is obtained when averaging over a sufficiently large number of transitions, i.e., when exploring decays fromEi >6.4 MeV where the level density is larger than ≈100 levels/MeV.

Similar results are obtained for Ef =1.345 MeV: in this case, the sudden deviations from a ratio equal to 1 are seen atE_γ ≈5 MeV, again corresponding toE_i ≈6.4 MeV. For Ef =3.34 MeV, the number of states in the final excitation energy bin is larger (>10 levels/MeV) and a ratio closer to 1 is observed all over the measured γ-ray energy range.

This is in agreement with the results from Ref. [15], which showed that for decays within the quasicontinuum the results were consistent with an excitation-energy-independent γSF forEf >3 MeV.

IV. SUMMARY

Particle-γ coincidence data from Refs. [19,20] was further analyzed to study the statistical behavior of theγSFs of⁶⁴Ni and⁶⁵Ni. The dependence of theγSF on both initial and final excitation energies was studied. In other words, the validity of the gBA hypothesis was investigated for the present case. In addition, the role of Porter-Thomas fluctuations as a function of excitation andγ-ray energies was analyzed.

The results suggest that theγSFs of^64,65Ni are independent of the initial excitation energy and in agreement with theγSF

from Refs. [19,20] at initial excitation energies of 5.82–9.66 and 4.43–6.08 MeV and at γ-ray energies above 2.0 and 1.6 MeV, respectively. TheγSFs corresponding to the feeding of the ground and first excited states present large deviations from theγSFs obtained with the standard Oslo method unless decays from sufficiently high initial excitation energies are considered, here above ≈5.8–6.4 MeV for ⁶⁴Ni and above

≈4.4 MeV for ⁶⁵Ni. These deviations, scattered around the excitation-energy-independentγSF, can be attributed to large Porter-Thomas fluctuations and probably also to the admix- ture of quadrupole transitions. With the exception of these deviations, the strength feeding a given final excitation energy bin is in good agreement with the assumption of a common γSF given by the standard Oslo method for all the excitation and γ-ray energies considered. Therefore, the results presented in this work support the validity of the generalized Brink-Axel hypothesis in^64,65Ni, assumed in Refs. [19,20].

ACKNOWLEDGMENTS

We would like to thank INFN Laboratory Nazionali di Legnaro for providing the ⁶⁴Ni target. We are also grateful for the financial support received from the Research Council of Norway (NFR). S.S. and G.M.T. acknowledge funding under NFR project Grants No. 210007 and No. 262952/F20.

A.C.L. acknowledges financial support from the ERC-STG- 2014 under Grant No. 637686. Finally, we would like to thank J. C. Müller, A. Semchenkov, and J. Wikne for providing the beam for our experiments.

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